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A cover page for a closed-book examination in differential equations at the university of british columbia (ubc) from april 2010. The examination includes instructions, rules, and 5 problem sets covering topics such as fourier series, boundary value problems, phase plane analysis, and laplace transforms.
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The University of British Columbia Sessional Examinations - April 2010
Mathematics 256 Differential Equations
Closed book examination Time: 2 12 hours
Name Signature
Student Number Instructor’s Name
Section Number
One 8 12 ” × 11” two sided cheat sheets are permitted. One 8 12 ” × 11” one sided table of Laplace transforms is permitted. Non-programmable calculators allowed. Show your work in the spaces provided. You are encouraged to explain all steps in your solutions.
Rules Governing Formal Examinations
Total 100
Marks
[20] 1. The function f (x) is defined for 0 ≤ x ≤ 2 by f (x) = x(2 − x). (a) Sketch the odd and even extensions to f (x) over the range − 2 ≤ x ≤ 2. Would you expect the Fourier sine series to converge to f (x) faster or slower than the Fourier cosine series? Explain your reasoning in 1-2 sentences.
(b) Compute the Fourier sine series to f (x).
Marks
[20] 2. (a) Find the general solution of the following homogeneous linear system:
x′^ =
x,
and sketch the phase plane close to x = 0. Is this point a saddle, a node, a spiral or a centre?
(b) Find the solution of the following initial value problem:
x′^ =
x +
−e−^2 t − 6 e−^2 t
, x(0) =
What is unusual about the initial conditions?
(c) Express the steady state response from part (b) in phase-amplitude form: yp(t) = R cos(ωt − φ 0 ), showing that the amplitude R depends only on ω^2. What value of ω within the range 1 ≤ ω ≤ 5 will maximize the amplitude?
[20] 4. Consider the the following initial boundary value problem
ut = uxx − 2 , 0 < x < 2 , t ≥ 0 ,
subject to the boundary and initial conditions:
u(0, t) = 1, u(2, t) = 1, u(x, 0) = 2
(a) Writing u(x, t) = us(x) + v(x, t) state the problems that are satisfied by the steady state solution, us(x), and the transient solution v(x, t).
(b) Find the steady state solution, us(x).
[20] 5. (a) Find the Laplace transform of f (t) = t sin t.
(b) Find the inverse Laplace transform of
F (s) =
5 s + 10 s^2 + s − 6
Solve the following initial value problems using Laplace transforms and describe the behaviour of y(t) as t → ∞:
(c) y′′^ + 8y′^ + 12y = 0, y(0) = 1, y′(0) = 0
.
(d) y′′^ + 8y′^ + 12y = δ(t − 2) + 2u 1 (t) − u 3 (t), y(0) = 0, y′(0) = 0
.
The End